I'm having a tough time visualizing the whole system of state plane grid. You have a perfectly flat, square grid of coordinates, in some kind of relation to an ellipsoid. You get rays from the center of the ellipsoid projected at some angle through this grid and where that hits is your coordinate.
It seems to me that the spatial relationship between these two elements would effect the coordinates - the further away from the ellipsoid the grid is, the longer the distance between the planar intersections of the two rays.
What's never been explained to me is, where/how high or low is this grid to the ellipsoid? You always seen them drawn together, but is this plane tangent at some point, or where the heck is it?
I think it depends on the type of projection, but I believe in the Mercator projection the cylinder is tangent to the ellipsoid at the central meridian of the zone. (I guess maybe a central latitude if it's transverse).
Take a look at the PDF for Class 5 at my archive site: http://geodesyattamucc.pbworks.com (or click the house symbol next to my name).
Many details. Many pictures. Hopefully, the answers you need.
In almost ALL cases, the "plane" is inside of the ellipsoid. I believe American Samoa has a tangent case; don't remember about the Virgin Islands. For Lambert Conformal Conic, for Transverse Mercator, and for Oblique Mercator the scale factor at origin is less than unity - it's inside of the ellipsoid at the origin with a scale factor (Grid to Ground) of around one part in ten thousand, more or less.
It is secant, intersecting at the lines of incidence.
If I recall correctly, this guy gives a pretty good discussion:
http://www.dot.ca.gov/hq/row/landsurveys/LSITWorkbook/WorkbookTOC.html
That sounds more right to me than what I had said before. I don't have my books with me here, but I remember learning it as you have described.
In Denver, it's pretty High!
🙂
http://cdm.sos.state.ga.us:2011/cdm/ref/collection/fieldnotes/id/7078
As my friend the Section 4, 10th District Surveyor from 1832 said "As High as could be wished"
[sarcasm]For the most part, it ain't sea level.[/sarcasm]
Nate,
Wouldn't the state plane grid be really low in Denver?
Cause it is the mile high city. And, now, it has moved to be 2-3 miles high!
(Pot was legalized, just for Ted, but that's another story!)
N
Okay...maybe I shouldn't take this conversation further....but.... in that case, wouldn't the "grid" be 2 to 3 miles below the "Highness" of Denver?
I'm not so sure the "grid" has a height
Does the "grid" really have a height? I don't think so because the coordinate (this is just an example) 100,000N, 100,000E can be at any height. Where you are within the grid makes a difference for the scale factor - there are 2 lines of unity and between the unity lines the scale factor is negative (if I remember correctly - but I may have it backwards) and outside the unity lines the scale factor is positive. Then there is the height correction factor.
So there is no place that is above or below the grid because the grid is at any height.
Hope that makes some sense as to what I'm saying. And, more importantly, I hope I have my head on straight.
It's a secant plane, so it intersects the ellipsoid along two lines and its depth below the ellipsoid varies. For a Lambert Projection, its maximum depth below the ellipsoid is at the central parallel. You can calculate that number from the tables given in Stem's Manual NGS 5. The two zones for North Dakota are found on page 101.
The last number in the table is little r sub zero (r0). It is the mean radius of the ellipsoid at the central parallel, scaled to the plane. That is, r0 is the mean radius multiplied by the scale factor at the central parallel, which is k0 in the table.
To calculate the maximum depth of the state plane, you need to subtract the scaled mean radius from the mean radius. You can calculate the mean radius by dividing the scaled radius by the scale factor.
So, the calculation goes like this: Max Depth = r0/k0 - r0. For Zone 3301, North Dakota North, Max Depth = 6,379,995/.999935842096 - 6,379,995 = 409.353 meters.
> In almost ALL cases, the "plane" is inside of the ellipsoid.
Secant Lambert projections are only inside the ellipsoid in-between the standard parallels. North and south of these, the grid is outside of the ellipsoid.
Not sure about the rest of the US, but all of California's state planes systems are these secant lambert projections.
I'm not so sure the "grid" has a height
I believe Dave is on to something here. Consider:
1. There is a difference between a graphical projection and a mathematical projection.
2. State plane coordinate systems are conformal which is a mathematical condition.
3. The Cauchy-Riemann equations used to establish the condition of being conformal have no variables in the third dimension.
4. Therefore, strictly speaking, there is no definition of vertical in a map projection that can be tied to elevation.
From a conceptual visualization perspective, the graphical diagrams are very helpful.
On the other hand, the global spatial data model (GSDM) preserves the 3-D integrity of geospatial data. The GSDM also supports all the advantages of a low distortion projection (LDP) in addition to providing convenient methods for handling error propagation and spatial data accuracy.
I'm not so sure the "grid" has a height
Well put, but I hope that you would agree that a state plane is a fixed surface secant to the ellipsoid. That being so, there is a separation; i.e. a distance from the plane to the ellipsoid at every point within the plane's useful domain, ranging from negative between the secant intersections to 0 at those intersections to positive outside those points.
I took the original question to ask what that distance is. It has nothing to do with ellipsoidal height, which is the height of a point on the earth's surface above the ellipsoidal surface. Height refers to a point, but the question was about the plane.
Calculating that separation at the state plane's natural origin is easy, but it's problematic at other points. That's because a normal to the ellipsoid at other points is not perpendicular to the state plane, so choosing the definition of a perpendicular distance is troublesome.
I'm not so sure the "grid" has a height
Easy way around that problem is to approximate it with a Local Space Rectangular (LSR) projection from geocentric coordinates. For a typical Lambert State Plane Zone, compute the ellipsoidal separation at the projection's natural origin and then use that as the negative ellipsoid height to define an LSR. Once the coefficients are developed, it's easy to inverse from the plane back up to the ellipsoid surface to solve for the perpendicular distances. North-South from the origin is pretty consistent with what it is for a Lambert Conformal Conic, the logic deteriorates East-West, though ... probably good enough for a couple hundred kilometers.
I'm not so sure the "grid" has a height
Calculating the coefficients might be a problem, but the concept seems sound. A quick and dirty approach is to scale the normal to the point by the point's scale factor and subtract as you would at the natural origin. For a Lambert projection, the resulting segment is perpendicular to the ellipsoid, but not to the plane.
To get the segment perpendicular to the plane, raise the plane until the point in question is a secant point. Then the distance is calculated along the normal at the natural origin using the scale factor for the raised plane. I don't have a good way to do the calculation precisely, but I imagine that someone else does.
In any event, the difference between the two perpendicular distances is likely to be small.